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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク combination ： ウィキペディア英語版 combination In mathematics, a combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In smaller cases it is possible to count the number of combinations. For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a ''k''-combination of a set ''S'' is a subset of ''k'' distinct elements of ''S''. If the set has ''n'' elements, the number of ''k''-combinations is equal to the binomial coefficient :$\backslash binom\; nk\; =\; \backslash frac,$ which can be written using factorials as $\backslash frac$ whenever $k\backslash leq\; n$, and which is zero when $k>n$. The set of all ''k''-combinations of a set ''S'' is sometimes denoted by $\backslash binom\; Sk\backslash ,$. Combinations refer to the combination of ''n'' things taken ''k'' at a time without repetition. To refer to combinations in which repetition is allowed, the terms ''k''-selection,〔 also referred to as an ''unordered selection''.〕 ''k''-multiset, or ''k''-combination with repetition are often used.〔When the term ''combination'' is used to refer to either situation (as in ) care must be taken to clarify whether sets or multisets are being discussed.〕 If, in the above example, it was possible to have two of any one kind of fruit there would be 3 more 2-selections: one with two apples, one with two oranges, and one with two pears. Although the set of three fruits was small enough to write a complete list of combinations, with large sets this becomes impractical. For example, a poker hand can be described as a 5-combination (''k'' = 5) of cards from a 52 card deck (''n'' = 52). The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960. == Number of ''k''-combinations == (詳細はnumber of ''k''-combinations from a given set ''S'' of ''n'' elements is often denoted in elementary combinatorics texts by $C(n,k)$, or by a variation such as $C^n\_k$, $$ or even $C\_n^k$ (the latter form was standard in French, Romanian, Russian, Chinese and Polish texts). The same number however occurs in many other mathematical contexts, where it is denoted by $\backslash tbinom\; nk$ (often read as "n choose k"); notably it occurs as a coefficient in the binomial formula, hence its name binomial coefficient. One can define $\backslash tbinom\; nk$ for all natural numbers ''k'' at once by the relation :$\backslash textstyle(1+X)^n=\backslash sum\_\backslash binom\; nk\; X^k,$ from which it is clear that $\backslash tbinom\; n0=\backslash tbinom\; nn=1$ and $\backslash tbinom\; nk=0$ for ''k'' > ''n''. To see that these coefficients count ''k''-combinations from ''S'', one can first consider a collection of ''n'' distinct variables ''X''''s'' labeled by the elements ''s'' of ''S'', and expand the product over all elements of ''S'': :$\backslash textstyle\backslash prod\_(1+X\_s);$ it has 2''n'' distinct terms corresponding to all the subsets of ''S'', each subset giving the product of the corresponding variables ''X''''s''. Now setting all of the ''X''''s'' equal to the unlabeled variable ''X'', so that the product becomes , the term for each ''k''-combination from ''S'' becomes ''X''''k'', so that the coefficient of that power in the result equals the number of such ''k''-combinations. Binomial coefficients can be computed explicitly in various ways. To get all of them for the expansions up to , one can use (in addition to the basic cases already given) the recursion relation : which follows from =; this leads to the construction of Pascal's triangle. For determining an individual binomial coefficient, it is more practical to use the formula :$\backslash binom\; nk\; =\; \backslash frac.$ The numerator gives the number of ''k''-permutations of ''n'', i.e., of sequences of ''k'' distinct elements of ''S'', while the denominator gives the number of such ''k''-permutations that give the same ''k''-combination when the order is ignored. When ''k'' exceeds ''n''/2, the above formula contains factors common to the numerator and the denominator, and canceling them out gives the relation :$\backslash binom\; nk\; =\; \backslash binom\; n,\backslash text0\; \backslash le\; k\; \backslash le\; n.$ This expresses a symmetry that is evident from the binomial formula, and can also be understood in terms of ''k''-combinations by taking the complement of such a combination, which is an -combination. Finally there is a formula which exhibits this symmetry directly, and has the merit of being easy to remember: :$\backslash binom\; nk\; =\; \backslash frac,$ where ''n''! denotes the factorial of ''n''. It is obtained from the previous formula by multiplying denominator and numerator by !, so it is certainly inferior as a method of computation to that formula. The last formula can be understood directly, by considering the ''n''! permutations of all the elements of ''S''. Each such permutation gives a ''k''-combination by selecting its first ''k'' elements. There are many duplicate selections: any combined permutation of the first ''k'' elements among each other, and of the final (''n'' − ''k'') elements among each other produces the same combination; this explains the division in the formula. From the above formulas follow relations between adjacent numbers in Pascal's triangle in all three directions: :$\backslash binom\; nk\; =\; \backslash binom\; n\; \backslash frac\; k,\backslash textk>0$, :$\backslash binom\; nk\; =\; \backslash binom\; k\; \backslash frac\; n,\backslash text$, :$\backslash binom\; nk\; =\; \backslash binom\; \backslash frac\; nk,\backslash textn,k>0$. Together with the basic cases $\backslash tbinom\; n0=1=\backslash tbinom\; nn$, these allow successive computation of respectively all numbers of combinations from the same set (a row in Pascal's triangle), of ''k''-combinations of sets of growing sizes, and of combinations with a complement of fixed size . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「combination」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.